AI Tutor
Imagine two high school track teams sprint 100 meters, and you need to know which team is consistently faster and which has more spread in times. A standard histogram makes comparing specific percentiles awkward, but a cumulative curve reveals the entire ranking structure in a single glance.
How do we build this curve, and why does its steepness tell us so much about variability?

Cumulative Frequency and Boundaries

The cumulative frequency of a data class is the running total of frequencies up to and including that class. An upper class boundary is the exact value that separates one class interval from the next without gaps.
We always plot cumulative frequency against the upper class boundary because, by definition, all observations within a class interval fall at or below that upper edge.
What happens when we plot these points and connect them on a continuous coordinate plane?

Constructing an Ogive

In 1875, statistician Francis Galton borrowed the architectural word ogive—describing a Gothic pointed S-shaped arch—to name this smooth cumulative frequency curve.
To construct an ogive, anchor the curve at CF=0CF = 0 at the lower boundary of the first class, plot each point (upper boundary,CF)(\text{upper boundary}, CF), and connect them with a smooth, non-decreasing curve.
Now that the full distribution is mapped, how do we extract the median and spread?

Reading Medians, Quartiles, and IQR